Definition. Given a function $f$ defined on $[a,b]$, let $$\xi_k \in [x_{k-1},x_k],\quad k=1,\ldots,n$$ where $$ x_k=a+k\frac{b-a}n, \quad k=0,\ldots,n \; .$$
One says that $f$ is integrable on $[a,b]$ if the limit $$\lim_{n\to\infty}\frac{b-a}n\sum_{k=1}^n f(\xi_k)$$
exists and is independent of the $\xi_k$.

I seek a proof of the:

Theorem. If $a<c<b$ and $f$ is integrable on $[a,c]$ and $[c,b]$, then $f$ is integrable on $[a,b].$

Your version of integrability is equivalent to the standard one but (to me at least) seems rather awkward to work with. Thus I would begin by verifying that your definition implies that $f$ is Darboux Integrable: use Darboux's Integrability Criterion (Theorem 8.7 of these notes) and notice that by suitable choices of $\xi_k$'s you can make your sums arbitrarily close to the upper and lower sums.

Your theorem is one of the basic properties of the Darboux integral, so is proved in all the standard texts (see e.g. Theorem 8.8 of loc. cit.).